admin2/openfreebooks-bookgenerator
1
0
Fork 0
mirror of https://github.com/HananoshikaYomaru/openfreebooks.git synced 2026-11-13 13:05:51 +00:00
Code Issues Projects Releases Packages Wiki Activity Actions

Enhance chapter content and metadata: Update SKILL.md to include requirements for chart axes in all widgets. Implement axis labels, arrowheads, and graduations in the simultaneous graph explorer widget. Update math curriculum JSON to reflect the status change of multiple chapters to live. Add last modified timestamps to chapter templates and generate corresponding metadata for improved site navigation and SEO.

Browse source
This commit is contained in:
Hananoshika Yomaru 2026-05-21 00:33:01 -07:00
parent 353b129533
commit eee5d109d5
31 changed files with 3093 additions and 17 deletions
Download patch file Download diff file Expand all files Collapse all files

169
content/math/sequences-series/_index.md Normal file
View file

@ -0,0 +1,169 @@
+++
title = "Sequences & series"
description = "Work with arithmetic and geometric sequences, sigma notation, and finite summation formulas."
template = "chapter.html"
weight = 17
[extra]
subject = "math"
chapter_id = "sequences-series"
strand = "Number & Algebra"
+++
<div class="book-prose">
<p>
A <strong>sequence</strong> is an ordered list of numbers. A <strong>series</strong> is the sum of
terms in a sequence. These ideas are key in algebra, finance, and modeling.
</p>
<h2 id="sequence-notation" class="book-prose__heading">Sequence notation</h2>
<p>
Terms are often written \(u_1,u_2,u_3,\dots\) where \(u_n\) is the \(n\)-th term.
</p>
<p><strong>Example:</strong> \(2,5,8,11,\dots\) has \(u_1=2\), \(u_2=5\), \(u_3=8\).</p>
<h2 id="arithmetic-sequences" class="book-prose__heading">Arithmetic sequences</h2>
<p>
Difference between consecutive terms is constant \(d\).
</p>
<p class="book-formula">\[
u_n=a+(n-1)d
\]</p>
<p>
where \(a\) is first term.
</p>
<p><strong>Example:</strong> \(a=4,d=3\Rightarrow u_n=4+3(n-1)=3n+1\).</p>
<h2 id="arithmetic-series" class="book-prose__heading">Arithmetic series</h2>
<p>
Sum of first \(n\) terms:
</p>
<p class="book-formula">\[
S_n=\frac{n}{2}\big(2a+(n-1)d\big)=\frac{n}{2}(a+l)
\]</p>
<p>
where \(l\) is the \(n\)-th term.
</p>
<h2 id="geometric-sequences" class="book-prose__heading">Geometric sequences</h2>
<p>
Ratio between consecutive terms is constant \(r\).
</p>
<p class="book-formula">\[
u_n=ar^{n-1}
\]</p>
<p><strong>Example:</strong> \(3,6,12,24,\dots\) has \(a=3,r=2\).</p>
<h2 id="geometric-series" class="book-prose__heading">Geometric series</h2>
<p>
For \(r\ne1\), sum of first \(n\) terms:
</p>
<p class="book-formula">\[
S_n=a\frac{1-r^n}{1-r}
\]</p>
<p>
Equivalent form: \(S_n=a\frac{r^n-1}{r-1}\).
</p>
<h2 id="infinite-geometric" class="book-prose__heading">Infinite geometric series</h2>
<p>
If \(|r|&lt;1\), the sum to infinity exists:
</p>
<p class="book-formula">\[
S_\infty=\frac{a}{1-r}.
\]</p>
<p>
If \(|r|\ge1\), no finite sum to infinity.
</p>
<h2 id="sigma-notation" class="book-prose__heading">Sigma notation</h2>
<p>
\[
\sum_{k=1}^{n} f(k)
\]
means add values of \(f(k)\) from \(k=1\) to \(k=n\).
</p>
<p><strong>Example:</strong> \(\sum_{k=1}^{4}(2k+1)=3+5+7+9=24.\)</p>
<h2 id="problem-solving" class="book-prose__heading">Typical problem-solving workflow</h2>
<ol>
<li>Identify arithmetic or geometric behavior.</li>
<li>Find key parameters (\(a,d\) or \(a,r\)).</li>
<li>Use \(u_n\) for term questions, \(S_n\) for sum questions.</li>
<li>Check reasonableness (growth/decay, sign, size).</li>
</ol>
<h2 id="exam-strategy" class="book-prose__heading">Exam strategy</h2>
<ul>
<li>Do not mix \(u_n\) and \(S_n\) formulas.</li>
<li>For geometric sums, check \(r\ne1\) before applying formula.</li>
<li>For infinity sums, always test \(|r|&lt;1\).</li>
<li>Define variables clearly before solving word problems.</li>
</ul>
</div>
<div class="book-prose">
<h2 id="history" class="book-prose__heading">History</h2>
<p>
Sequences and sums appear in ancient mathematics, including arithmetic progressions used for
counting and trade. Later, geometric series became central in finance and calculus.
</p>
<h2 id="derivation" class="book-prose__heading">Derivation and reasoning</h2>
<h3 id="derive-arithmetic-sum" class="book-prose__heading">Why the arithmetic sum formula works</h3>
<p>
Write \(S_n=a+(a+d)+\dots+l\) and reverse it. Adding line-by-line gives \(2S_n=n(a+l)\), so
\(S_n=\frac{n}{2}(a+l)\).
</p>
<h3 id="derive-geometric-sum" class="book-prose__heading">Why the geometric sum formula works</h3>
<p>
Let \(S_n=a+ar+\dots+ar^{n-1}\). Multiply by \(r\), subtract, and simplify:
\[
(1-r)S_n=a(1-r^n).
\]
</p>
<h2 id="checkpoints" class="book-prose__heading">Checkpoints</h2>
<details class="book-question">
<summary class="book-question__prompt">Find the 20th term of \(5,8,11,\dots\).</summary>
<div class="book-question__solution">
<p><strong>Answer:</strong> \(u_{20}=5+19(3)=62\).</p>
</div>
</details>
<details class="book-question">
<summary class="book-question__prompt">Find \(S_{30}\) for \(2,5,8,\dots\).</summary>
<div class="book-question__solution">
<p><strong>Answer:</strong> \(a=2,d=3\Rightarrow S_{30}=\frac{30}{2}[2(2)+29(3)]=1365\).</p>
</div>
</details>
<details class="book-question">
<summary class="book-question__prompt">For \(3,6,12,\dots\), find the 8th term.</summary>
<div class="book-question__solution">
<p><strong>Answer:</strong> \(u_8=3\cdot2^7=384\).</p>
</div>
</details>
<details class="book-question">
<summary class="book-question__prompt">Compute \(1+\frac12+\frac14+\frac18+\dots\).</summary>
<div class="book-question__solution">
<p><strong>Answer:</strong> \(a=1,r=\frac12\Rightarrow S_\infty=\frac{1}{1-1/2}=2\).</p>
</div>
</details>
<details class="book-question">
<summary class="book-question__prompt">Evaluate \(\sum_{k=1}^{5}(k+2)\).</summary>
<div class="book-question__solution">
<p><strong>Answer:</strong> \(3+4+5+6+7=25\).</p>
</div>
</details>
<h2 id="applications" class="book-prose__heading">Applications</h2>
<ul>
<li><strong>Savings plans:</strong> recurring deposits form sequences and cumulative sums.</li>
<li><strong>Population models:</strong> repeated growth can be modeled geometrically.</li>
<li><strong>Computer science:</strong> loops and algorithm costs often use summation notation.</li>
</ul>
<h2 id="references" class="book-prose__heading">References</h2>
<ul>
<li>Standard high-school algebra curricula on arithmetic and geometric progressions.</li>
<li>Precalculus texts on sigma notation and finite/infinite geometric series.</li>
</ul>
</div>

91
content/math/sequences-series/core.html Normal file
View file

@ -0,0 +1,91 @@
<div class="book-prose">
<p>
A <strong>sequence</strong> is an ordered list of numbers. A <strong>series</strong> is the sum of
terms in a sequence. These ideas are key in algebra, finance, and modeling.
</p>
<h2 id="sequence-notation" class="book-prose__heading">Sequence notation</h2>
<p>
Terms are often written \(u_1,u_2,u_3,\dots\) where \(u_n\) is the \(n\)-th term.
</p>
<p><strong>Example:</strong> \(2,5,8,11,\dots\) has \(u_1=2\), \(u_2=5\), \(u_3=8\).</p>
<h2 id="arithmetic-sequences" class="book-prose__heading">Arithmetic sequences</h2>
<p>
Difference between consecutive terms is constant \(d\).
</p>
<p class="book-formula">\[
u_n=a+(n-1)d
\]</p>
<p>
where \(a\) is first term.
</p>
<p><strong>Example:</strong> \(a=4,d=3\Rightarrow u_n=4+3(n-1)=3n+1\).</p>
<h2 id="arithmetic-series" class="book-prose__heading">Arithmetic series</h2>
<p>
Sum of first \(n\) terms:
</p>
<p class="book-formula">\[
S_n=\frac{n}{2}\big(2a+(n-1)d\big)=\frac{n}{2}(a+l)
\]</p>
<p>
where \(l\) is the \(n\)-th term.
</p>
<h2 id="geometric-sequences" class="book-prose__heading">Geometric sequences</h2>
<p>
Ratio between consecutive terms is constant \(r\).
</p>
<p class="book-formula">\[
u_n=ar^{n-1}
\]</p>
<p><strong>Example:</strong> \(3,6,12,24,\dots\) has \(a=3,r=2\).</p>
<h2 id="geometric-series" class="book-prose__heading">Geometric series</h2>
<p>
For \(r\ne1\), sum of first \(n\) terms:
</p>
<p class="book-formula">\[
S_n=a\frac{1-r^n}{1-r}
\]</p>
<p>
Equivalent form: \(S_n=a\frac{r^n-1}{r-1}\).
</p>
<h2 id="infinite-geometric" class="book-prose__heading">Infinite geometric series</h2>
<p>
If \(|r|&lt;1\), the sum to infinity exists:
</p>
<p class="book-formula">\[
S_\infty=\frac{a}{1-r}.
\]</p>
<p>
If \(|r|\ge1\), no finite sum to infinity.
</p>
<h2 id="sigma-notation" class="book-prose__heading">Sigma notation</h2>
<p>
\[
\sum_{k=1}^{n} f(k)
\]
means add values of \(f(k)\) from \(k=1\) to \(k=n\).
</p>
<p><strong>Example:</strong> \(\sum_{k=1}^{4}(2k+1)=3+5+7+9=24.\)</p>
<h2 id="problem-solving" class="book-prose__heading">Typical problem-solving workflow</h2>
<ol>
<li>Identify arithmetic or geometric behavior.</li>
<li>Find key parameters (\(a,d\) or \(a,r\)).</li>
<li>Use \(u_n\) for term questions, \(S_n\) for sum questions.</li>
<li>Check reasonableness (growth/decay, sign, size).</li>
</ol>
<h2 id="exam-strategy" class="book-prose__heading">Exam strategy</h2>
<ul>
<li>Do not mix \(u_n\) and \(S_n\) formulas.</li>
<li>For geometric sums, check \(r\ne1\) before applying formula.</li>
<li>For infinity sums, always test \(|r|&lt;1\).</li>
<li>Define variables clearly before solving word problems.</li>
</ul>
</div>

66
content/math/sequences-series/supplement.html Normal file
View file

@ -0,0 +1,66 @@
<div class="book-prose">
<h2 id="history" class="book-prose__heading">History</h2>
<p>
Sequences and sums appear in ancient mathematics, including arithmetic progressions used for
counting and trade. Later, geometric series became central in finance and calculus.
</p>
<h2 id="derivation" class="book-prose__heading">Derivation and reasoning</h2>
<h3 id="derive-arithmetic-sum" class="book-prose__heading">Why the arithmetic sum formula works</h3>
<p>
Write \(S_n=a+(a+d)+\dots+l\) and reverse it. Adding line-by-line gives \(2S_n=n(a+l)\), so
\(S_n=\frac{n}{2}(a+l)\).
</p>
<h3 id="derive-geometric-sum" class="book-prose__heading">Why the geometric sum formula works</h3>
<p>
Let \(S_n=a+ar+\dots+ar^{n-1}\). Multiply by \(r\), subtract, and simplify:
\[
(1-r)S_n=a(1-r^n).
\]
</p>
<h2 id="checkpoints" class="book-prose__heading">Checkpoints</h2>
<details class="book-question">
<summary class="book-question__prompt">Find the 20th term of \(5,8,11,\dots\).</summary>
<div class="book-question__solution">
<p><strong>Answer:</strong> \(u_{20}=5+19(3)=62\).</p>
</div>
</details>
<details class="book-question">
<summary class="book-question__prompt">Find \(S_{30}\) for \(2,5,8,\dots\).</summary>
<div class="book-question__solution">
<p><strong>Answer:</strong> \(a=2,d=3\Rightarrow S_{30}=\frac{30}{2}[2(2)+29(3)]=1365\).</p>
</div>
</details>
<details class="book-question">
<summary class="book-question__prompt">For \(3,6,12,\dots\), find the 8th term.</summary>
<div class="book-question__solution">
<p><strong>Answer:</strong> \(u_8=3\cdot2^7=384\).</p>
</div>
</details>
<details class="book-question">
<summary class="book-question__prompt">Compute \(1+\frac12+\frac14+\frac18+\dots\).</summary>
<div class="book-question__solution">
<p><strong>Answer:</strong> \(a=1,r=\frac12\Rightarrow S_\infty=\frac{1}{1-1/2}=2\).</p>
</div>
</details>
<details class="book-question">
<summary class="book-question__prompt">Evaluate \(\sum_{k=1}^{5}(k+2)\).</summary>
<div class="book-question__solution">
<p><strong>Answer:</strong> \(3+4+5+6+7=25\).</p>
</div>
</details>
<h2 id="applications" class="book-prose__heading">Applications</h2>
<ul>
<li><strong>Savings plans:</strong> recurring deposits form sequences and cumulative sums.</li>
<li><strong>Population models:</strong> repeated growth can be modeled geometrically.</li>
<li><strong>Computer science:</strong> loops and algorithm costs often use summation notation.</li>
</ul>
<h2 id="references" class="book-prose__heading">References</h2>
<ul>
<li>Standard high-school algebra curricula on arithmetic and geometric progressions.</li>
<li>Precalculus texts on sigma notation and finite/infinite geometric series.</li>
</ul>
</div>